Conformal symmetry and the nonlinear Schrodinger equation

We show that the width of the wave-packet of a class of generalized nonlinear Schrodinger equations (NLSE) trapped in an arbitrary time-dependent harmonic well in any dimensions is universally determined by the same Hill's equation. This class of generalized NLSE is characterized by a dynamical O(2,1) symmetry in absence of the trap. As an application, we study the dynamical instabilities of the rotating as well as non-rotating Bose-Einstein condensates in one and two dimensions. We also show exact extended parametric resonance in a non-relativistic Chern-Simons theory producing a gauged NLSE.